Question

If ${\rm{A}}$ is a symmetric matrix, then matrix ${M^\prime }$AM is
A. Symmetric
B. Skew-symmetric
C. Hermitian
D. Skew-Hermitian

Answer 1

Hint: Here, we'll first utilize the formulas AT = A for symmetric matrices and AT = - A for skew-symmetric matrices, where the diagonal elements are also zero. Then we'll locate a matrix that accommodates everything. In our case, we have been given that ${\rm{A}}$ is a symmetric matrix and are asked to find the matrix type of ${M^\prime }$AM for that we have to use the concept of symmetric matrix to determine the desired answer.

Formula Used: A square matrix B of dimension n \times n is thought to be symmetric if and only if
BT = B

Complete step by step solution: We have been provided in the question that,
We have to assume that {\rm{A}} as a symmetric matrix
And we are to determine what matrix is ${M^\prime }$AM
We have been already known that,
A square matrix that is identical to the inverse of its transpose matrix is said to be skew symmetric. A square matrix, which is the same as its conjugate transpose matrix, is a Hermitian matrix.
From the given information, we came to a conclusion that
$\left( {M'AM} \right)' = M'AM$
Since, A is symmetric then M'AM is a symmetric matrix.
Therefore, if ${\rm{A}}$ is a symmetric matrix, then matrix ${M^\prime }$AM is a symmetric matrix.

Option ‘A’ is correct

Note: Students should be aware that the skew-symmetric matrix is a square matrix that equals the negative of its transpose and that the symmetric matrix is a square matrix that equals its transpose while addressing issues of this nature. A transpose of a matrix is an operator that flips a matrix over its diagonal, where it produces another matrix, represented as A' by switching the row and column indices of the matrix A.